Perform the multiplication and use the fundamental identities to simplify.
(cotx + cscx)(cotx-cscx)
I know that you have to foil first so
cot^2x - csc^2x and then use the pythagorean identity 1+cot^2u = csc^2u but I do not know how to simplify.
You have the solution right there in front of you ...
cot^2x - csc^2x
= cot^2x - (1 + cot^2x)
= cot^2x - 1 - cot^2x
= - 1
To simplify the expression (cotx + cscx)(cotx - cscx), you are correct that you need to use the FOIL method first. FOIL stands for First, Outer, Inner, Last, and it helps you multiply two binomials.
Using the FOIL method, you multiply the terms in the first parentheses with the terms in the second parentheses:
(cotx + cscx)(cotx - cscx) = cotx * cotx + cotx * (-cscx) + cscx * cotx + cscx * (-cscx)
Simplifying this expression gives you:
cot^2x - cotx * cscx + cotx * cscx - csc^2x
Notice that cotx * (-cscx) and cscx * cotx are the same, so they cancel each other out:
cot^2x - cotx * cscx + cotx * cscx - csc^2x = cot^2x - csc^2x
Now, to further simplify this expression, you can use the fundamental identities. One such identity is the Pythagorean Identity for trigonometric functions, which states:
1 + cot^2x = csc^2x
Rearranging this identity, you have:
cot^2x = csc^2x - 1
Substituting this into the expression cot^2x - csc^2x, you get:
(csc^2x - 1) - csc^2x = csc^2x - 1 - csc^2x = -1
Therefore, the simplified expression is -1.